The information on the pretty website is spare. However, there are a few issues which give me pause.
The integral example already given is one. One must wonder what is going on "under the hood" to give such a result. My hypothesis is that Riemann sums are what's going on, and that the integral uses 0 as a sample point. An interesting point is that this is one of the unit tests. https://github.com/gogotanaka/Hilbert/blob/master/test/inter...
Another is the example of lim[x->0] 1/x => oo. This is not true; the limit does not exist. If one said lim[x->0+] => oo there would be a better case. Also a unit test. https://github.com/gogotanaka/Hilbert/blob/master/test/inter...
(1 2 3; 4 5 6) * (1 2 3) => (14 32) is not how matrix multiplication is normally done. ( (1 2 3) would need to be (1; 2; 3).)
All in all, the grammar seems to be quite irregular; it seems to be the result of a sequence of wanting to solve one very particular kind of problem (take a limit, differentiate a function, take a sum) and cludging it into the parser.
https://github.com/gogotanaka/Hilbert/blob/master/lib/hilber... has seven exclamation marks in a row. That's amusing... I don't know anything about Ruby, but I feel that it must be a joke.